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Ihrig, Elizabeth A., Ed.Introducing Multiplication and Division, Kangaroosand Numbers: MINNEMAST Coordinated Mathematics -'Science Series, Unit 17.Minnesota Univ., Minneapolis. Minnesota SchoolMathematics and Science Center..National Science Foundation, Washington, D.C.71

70p.; For related documents/see SE021201-234;Photographs may not reproduce wellMINNEMAST, Minnemath Center,_720 Washington Ave.,S.E., Minneapolis, MN 55 4-,

u

EDRS PRICE MP-$0.83 HC-$3.50 Pies Postage.DESCRIPTORS *Curriculum Guides; Division; Elementary Education;

*Elementary School Mathematics; *Elementary SchoolScience; Experimental,Curridulum; *InterdisciplinaryApproach; learning Activities; Mathematics Education;Multitaioation; *Number Systems; Primary Grades;Process Education; Science Education; Units .of Study(Subject Fields); Whole Numbers

IDENTIFIERS *MINNEMAST; *Minnesota Mathethatics and ScienceTeaching Project

ABSTRACT, This volame is the seventeenth in a series of 29

coordinated MINNEMAST units in mathematics and science fdr' kindergarten and the primary grades. Intended for use by second -grade-teachers, this unit guide provides summary and 'overview of theunit, a list of materials needed, and descriptions of five groups of

,activities.'The purposes and procedures for each activity are` di-Stussed. Examples of questions and disbussion topics are given, 'and

in several cases ditto masters, stories for.reading aloud, and otherinstructional materials are included in the. book. In this unit,,multiplication is approached as repeated addition on a number line. .

In a second set of lessons, Multiplication is considered inconjunction with arrays. Addition and multiplication are thencompared, and simple fractions are introduced. A final review sectionis also included. (SD)

***********************************************************************Documents acquired by ERIC include many informal unpublished

* materials not available from other, sources. EPIC makes every effort ** to obtain the best copy available. Neverthelless, items of marginal ** reproducibility are often encountered and this affects the quality ** of the microfiche and hardcOpy reproductions ERIC makes available **s via the ERIC Document Reproduction Service (EDRS) . EDRS is not* responsible for the quality of the original document. Reproductions *,* supplied by EDRS are the best that can be made from the original.***********************************************************************

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KANTGARCICOS'& NUMBERS

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'MINNESOTA.- .MATHEMATICS AND SCiENCEEACHiN0.PROJECT:

MINNEMAST

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COORDINATED MATHEMATICS SCIENCE SERIES

-1.9

WATCHING AND WONDERING .- .

2. CURVES AND SHAPES

3. DESCRIBING AND CLASSIFYING

4. USING OUR SENSES..

5. INTRODUCING MEASUREMENT

6. NUMERATION4.

7. INTRODUCING SYMMETRY

8. OBSERVING PROPERTIES

9. NUMBERS AND COUNTING

10. DESCRIBING LOCATIONS

I I. INTRODUCING ADDITION ANDSUBTRACTION

12. MEASUREMENT WITH REFERENCE UNITS

13. INTERPRETATIONS OF ADDITION AND SUBTRACTION

14. EXPLORING SYMMETRICAL PATTERNS

15. INVESTIGATING SYSTEMS

16. NUMBERS AND MEASURING

17. INTRODUCING MULTIPLICATION AND DIVISION

18. SCALING ANDREPRESENTATION

19. COMPARING CHANGES

20. USING LARGER NUMBERS

21. ANGLES AND SPACE

22. PARTS AND PIECES

23. CONDITIONS AFFECTING LIFE

24. CHANGE AND CALCULATIONS

MULTIPLICATION ANDMOTION

26. WHAT ARE THINGS MADE OF?

27. NUMBERS AND THEIR PROPERTIES'.

28. MAPPING THE GLOBE

29. NATURAL SYSTEMS

Ct

OTHER MINNEMAST PUBLICATIONS -

The 29 coordinated units and several gther publications are available from MINNEMAST on order.Other publications include:

'STUI5ENT MANUALS for Grades I , 2 and 3, and.>printed TEACHING AIDS for Kindergarten and Grade I .

LIVING THINGS IN FIELD AND CLASSROOM(MINNEMAST Handbook for all grades)

ADVENTURES IN SCIENCE AND MATH(Historical stories for teacher or student)

'qUESTIONg AND ANSWERS ABOUT MINNEMASTSent free with price list on request

OVERVIEW(Description of content of each publication)

MINNEMAST RECOMMENDATIONS FOR SCIENCE AND MATH IN THE INTERMEDIATE GRAD(Suggestions for programs to succeed the MINNEMAST Curriculum in Grades 4, 5 and 6)

c.)

INTRODUCINGMULTIPLICATIONAND DIVISIONKANGAROOS

ISTVIVIBERSA

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UNIT

.4,

MINNESOTA MATHEMATICS AND SCIENCE TEACHING PROJECT720 Washi gton Avenue .S: E. , Minneapolis., Minnes6ia 55455

'

eM N N E

DIRECTOR JAMES WERNTZ,Professor of PhysicsUniversity of Minnesota

'ASSOCIATE DIRECTORFOR SCIENCE

AS-S0CfATE'DIRECTOR40FOR-RESE4RCH 4ND EVALUATION

r,4.

ROGER S. JONES"Associate Professor of PhysicsUniversity of-Minnesota

WELLS HIVELYAssociate Professor -of PsychologyUniversity of MinnesOta

c

.

Except for the rights to materials reserved by others , the publisherand .copyright owner hereby grants permission to domestic persons ofthe linited-Sates and Canada for use of this work without charge inEngliSh language publications in the United. States and Canada after

-July I , 1973.,; provided the publications incorpOrating materials coyereti°. by these copyrights. contain acknowledgement of them and a statementthat.,the .publication is endorsed neither by the copyright owner nor theNatiOnal Soience,Foundation. For conditions of use and permission to

,, use material's contained herein for foreign publications or publicationsin other than the English language, application must be made to: Officeof the University Attorney, University of Minnesota, Minneapolis,Minngsota 55455:

'

The. Minnesota Mathematics and Science Teaching Project .

developed these materials under a grant from the ,.National Science 'Foundation, . -,

196.8, Univergity of Minnesota. All rights reserved.

Third Printing , 1971'

5

AND EavisxortI ,4

% -This-pnit-was-develop-ed-bS,:MINNEMAST on the,..'basis-of expeyienc&-s of.thp many teachers whotaught. an, earlier version in. th.9,ir classrows.

.-... tr!.1".

. .. . .

r ,I o a 4- " Wt. 1l4 4

KAY W. ilLAIAssistant-Professor of MathematicsMinnesota School Matheinatics and Sciende Center

4

C. MURRAY BRADENProfessor of MathematicsMacelaster College

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EDMUND C.. BRAY

ELIZABETH*.A. TERIG

6K'KABAT.DITH L. NORMAN

-BARBARA A. MORRIS:'

6

.

,

. 4

Assistant to Director.,Curficul4m MaterialsDevelopment "

Editor

Art- DirectorAsSistant

Materials-Litt for Unit .

-. Introduction

Section I.

Lesson

.0

CONTENTS-.

Multiplication by Repeated Addition on a Number,. Line

Lesson 2

Lesson 3Lesson 4

Lesson 5.

Repeat61:JumpS on the Number LineStory: "Kangaroo Games"Multiplication. as Repeated. Addition on a

Number Uri-6-i'Showing Multiplication on diraliel. Number

Multiplication 'ChartsMaking a Multiplication Chart Using the

Number Line,e .

Section 2. Arrays, and Multiplicatioli

Lessons 6 ,

'Lesson 7

Lesson 8

"Lesson 9

Lesson 10

Multiplication Using Arfay's .

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1

4

5

5

12.Linis 15

18

Making ArraysCommutativity of MultiplicationMaking a MultiplicatiOn Chart Usiflg Arrays. .Areas of lectangleS with Integral Lengths

Section 3. Comparing Addition and Multiplication.. o k

Lesson IILesson 12Lesson 13

Making an Addition ChartComparing Addition and Milltipfication.

'Applying Multiplication and Addition

Section 4.i Sdme Fractions-

.Lesson 14 Some Simple FractionsStory: "fimmy's,Favorite Cookies"

Lesson" 15 Fractional Parts of Plane FiguresLesson" 16 I4lves on the Number Line

Section 5. Review

Kpngaroo Cuto Uts for LeSson 1

Fish Pattern for Lesson 8

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Complete List of Materials for Unit,...17

(Numbers bpsed on class size of 30.)

total numberrequired to itemteach unit

30 * *Student Manuals .

1 demonstratidn number line(optional)

30. *nufnber 0-2 5

750 * counters

.30 pegboards and pegs(can substitute counters).

2 'flannel boards .

(optional)

2-3 4sets of objects for flannel'board

24 fish Or other objects for flannel board

30 ***sets of Mirinebars

30.

***addition slide rules(if needed)

24 cookies or other objects for flannel board

scissors18.-inch

,strips of paper

*kit items as'wellas°**printed materials available from

Minnemath Center, 720 Washington Ave. S.E. ,'Mpls., Minn. 554 55

o

lesson's in. which item

is used

4, 5,

4,5,11

'6,7,10,11,14

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7

8

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11

14

15

16

-***available from The Judy Company.3 10 North Second Street, MinneaPolis, Minnesota 55401

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INTRbDUCTIC*

4.

NItiltiplicatioriis,a mathematical operation that yields from .

a,ny two real numbersta-uniqUe third number called the product.Multiplication of 3 an -4 .yields the product 3 x 4. "3 x 4"is a name of p nurnber that we also know as 12. It is also

-.possible. to" think of "Multiplitation by 3". as an operation that`thhisfortns 4 into the-product 3 x 4,

tirThe method used to find the single numeral. 12, which alson.i.nies the product 3 X 4, is called a\multiplicatiOn algorithm.There are several such algorithms available, Those that willbe used at some time in the MINNEMAST program are:

I. Repeated addition using jumps on a number line.2. Repeated addition using arrays or combinations of

equivalent sets. ,

a. Related scales on parallel number lines.4. Cartesian products. (This wi,10e explained in a,later

second grade 'unit.)5. A graphical method using aline of given slope..,6. A methorized table and computational rules.7. A sli4 rule.

These algorithms differ in their g.enerality. For example, algo-rithm 4 can be used only for the counting numbers 1, 2, 3,whereas,the graphical method'of algorithm 5 can be applied to .

all real number (including, e.g. r2)'. In this unit we chooseto use the first three algorithms. Some rdasons for this choiceare listed below. 4

All three algorithms have a strong connection with addition,with which the children are familiar.Each algorithm can be"given a simple physical interpretation..As the need arises' for multiplying other numbers such asfrictions, the, children need not unlearn the methods learned

3,

0

.3

4

.

. in this unit but can modify them

. All rules for computation cation, multiplication., and cli isdo

'5, 67, 8, and.9.4 . Children need simplejnethocis for ffnding,thsi exact.product

of tiny whote nif,mbers.that they do not remember. The\,first. three a lgorites listeckabovp provide thesse methods.

14,

as th6ykkpand their concepts.rreducecl to addition, subtrac-.n of the'numb'ers '0, I , 2, 3,

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Different algo4hms are .presented sc.) that the child will viewthe concept of rhultiplication various ways. If a child isslow to.t.ni iersiand one:interprdtation, 1.e May understand itbetter after he sees anothersinterprotation. Although the dif-ferent algorithms are clLisely rel&ec, the relations wit not bemade explicit. kjowever,sthe examples will -give the childrena feeling that underlyingill'of them is the-mathematical ideaof multiplication.

In this unit the Children will work with only the nwither'':0 l',.6 2, S., 4 5, and-6. We restrict ourselves -m='to these small nu-m='

1. bers.because they are easier for the'children to .Work with, and

at the same time they show the- methods as fully as larger num-bers would. k

2

4,

4SI

Why Introduced at this Time

You may be wondering Why simple multiplication is intr oducedso early in thessecond grade materials, before thd children- .have-had much opportunity to leaf i systematic procedures foraddition and subtractiOn. There area several reasons for ourearly consideration of multiplication.

First, the testing of this unit has shomiNthat second: graderscan easily grasp the sconcepts -presented .here. In fa'ct, mani,become familiar with simple multiplication even when it is notpresented to them in their school work.

Second, it is possible to give the children a much better un-derstapdingOf our positional system of nuMeratiOn if theyderstand products. For example, they will use the idea that"375" means (3 x 100) 1- (7 x 10) + 5. '

9

Third, it beco6.s possible to deal with many of the non-prith-%metical second grade topics m&e meaningfully if the childrenhave a concept of produdts. This is true of the material onscaling and representation, and on areas of rectangles.

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SECTION I IVIILLTIPPCATON BY REPEATED ADDITION ON A NUMBER LINE

this 'section it is shown'that the product of time countingmay be found by repeated lumps on a number line.

Twolbaratlel number lirtes differing inscale are used togive a picture of the result of multiplying by a counting num-ber. The concept of a-pair of scgled number lines is alsoused in Unit 18, Scaling ana':Representation. Later, pairsof scaled number lines are used for-division and for multi-..plication of numbers other than the counting. numbers.

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Lesson 'I: RCPCATCD JUMPS ON TILE NUMBER LINE'

Thjs les,spn treats repeated addition cif a counting Nmberby repeating jumps of equal length on the number line.Multiplication is not yet mentioned. The term "additionequation''isutrJd,t-liow-cveTT-you may use "addition sentence".Lf you wish.

MATERIALS-1"

Worksheets, , 5

PROCEDURE

Read the following.stopi t' the children as pn introduction tothe worksheets. You.inay"il-t. to Use an overhead projector .

or copy Worksheet I with pictes large enough to be seen byE 'the class when it is posted. There are kangargos to cu., out

at the end of the unit if you. wish "t use them- for demonstrations.

,KANGAROO GAMES

Australia is a large country and it has many wide-open spaceswhere few people live. One of these wide-open splices was chosenas a good spot for testing some new highway building machinery.Some men brought the equipment from far away.%They set up tents,unpa,gked their supplies, and.got the machines ready for testing the,

'next morning. When they finished all their work they scat resting on

camp chairs while the cook fixed their evening meal.

The men .thought they were in a deserted spot. But they had a.surprise waiting for therm.

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While they at and rested, 'a curious faMily kangarooS 'cameclose to watch them. There were a- father, a mother, a small babycalled Joey, and two. partly -grown kangaroos named Chuck andThe men and the'kangart es watched each other, until the cook calledthat supper was ready.

Wnile the men were eating, the kangaroos slowly hopped about,.

eating grass and othe.r.4)1ants. , Occassionally they took long jumps.

One of the men, Jim, said, "r like to watch these animals./How can we keep them near the camp?'

"I made too many biscuits tonight," answered' the cook. "Ifwe put the extra ones out near the kangaroos, 'perhaps they will eatthem and stay around hoping to get more."

pile.Bill put 3 biscuits in one pile, ^4 in another, and 2 in the-last."

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WHAT ADDITION EQUATION (or-addition sentence-) TELLS HOWMANY BISCUITS HE PUT OUT? (3 + 4 + .2 = 9)

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-The next day some of the men painted a number line on a longstraight road. The men 'planned to use the number linp in their machinetests, but the kangaroos thought it was::painted especially for them toplay on.

3- -The first evening after the ,nurnber line was painted the kanga-

roo family had jumrping contests along the line. They found thatFather 'Kangaroo jumped exactly 4 unit spaces each time. Mother

Kangaroo jumped 3 unit spaces each time. Chucl: made jumps of 2unit spaces, and Katie made jumps of only I unit space. Joey , thebaby of the family, couldn't jump at ally, so Mother Kangaroo carried'him wherever she went.

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DOING THE WORKS MEETS

...On Worksheet-I-the children are askea-to show on number linesfiow far each k-aniaroo'can go in one juihti. They may refer t6this workheet as they work on thee):ct feW.

On Worksheet 2 they will find on number lines the total num-ber of unit spaces covered by various combinations of kangaroojumps. 4 4

.. . .,.

On Worksheets 3, 4 and 5 the childreh will draW juinps and writeaddition sentences. You may \wish to discuss the first problemor two in 6 manner similar to the following:,

,. We can tell about the total', umber of unit spacesjumped. by: wriOng an, additio equation'. Lodi: at"Mother Kangaroo's trip on Worksheet 3. Who canwrite an addition equation 'telling about her trip?(3 + 3 +'3 + 3 = 1=2) . -We call addition such asthis, where one number is added, to its-elf severaltimes, repeated,addition."- \

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Color the Kangaroo jumps, `6.

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Katie.

I jump = I space

0 I 2. 3 4

No jump = 0 spaces

0 1 2 3 4

Joey

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Worksheet 2Unit 17 Name

'Fill in the boxes and show the jumps on the numberlines.

Father Kangaroo took 3 jumps,Each time he jumped 4 spaces.How many spaces did he jump altogether? lo?

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4

Chuck said to Katie, "I can go farther in 4 jumpsthan you can go In 7 jumps." Waslie right?

Remember: Chuck jumps 2 spaces at a time, andKatie jumps I space at a time.

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.NOTE: In the firs-f-problem on Work Sheet 5,Chisel; Kangaroo takes i"Wo---2,-.-_space jumps,re'sts, and then takes three 2-ifiace-,,-jumPs.The addition sentence describing this situi=--------tion,oan be written (2 + 2) + (2 + 2 71- 2) = 10,where the parentheses shbw the grouping ofthe jumps. Unless a child suggests this form,simply use 21- 2 + 2 -11 2 ÷ 2 = 10.

Worksheet 4'Unit 17 Name

Show the jumps on theboxes.

number lines and fill in the

Katie and Chuck were sitting on the zero mark, FatherCold them they could each jump. 4 times. Remember:Katie lumps I Space -at a time and Chuck jumps 2spades at a time.

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How far did Katie jump?

How far did Chug jump?

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An addition equation that tells about Katie's Jump is

spaces

spaces

An addition equation that tells about Chuck's Jump is

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Worksheet 3Unit 17 Name

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Show the jumps on the number lines and fill in ,thebokes.

Mother Kangaroo took 4 jumps of 3 unit spaces each

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How far did she go..? spa .esi4 5 6 7 8 9 10 I I 12

Write an addition equation that tells about her jumps.

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The next time her jumps looked like this:

0___,1 2 3 .1 5 6 Y 8 9 1 0 I I 12

Write an athliton equation.bout these, jumps.

Worksheet 5Unit 17 Name

Chuck Kangaroo jumps 2 spaces at a tune.He started at 0, took 2 jumps, rested, and thentook .3 more jump's."' va

On What number did he stop/ -;AO

1 1 l'..1 'I>0 1, 2 .3 5 6 7 8 .9 I,

Write an addition sentence about his lumps.

How far can Chuck go in .3 jumps') spaces

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Lesson 3: SHOWING MULTIPLICATION ON P'ARALL'EL NUMBER DINES .

Here a parallel number line method of showing,multiplicationis introduced. We will use the jumps of the Kangaroo familyto develop two parallel number lines of different scales. Thesepairg will be called a 2 -shale or a 3-scale or a 4-Scale', etcpIt is not intended' that the students learn the 'reaping of theword "parallel". It is important they" learn_ to read, the productsfrom the number lines,

'Similar parallel scaled, number lines are developed in Lesson 4'ofNUnit 18.

MATERIALS

- Worksheets .10 11 and 12

PROCEDURE

-

Draw a number line On the chalkboard. Have one' of yourstudents show Chuck Kangaroo's jumps on the number linestarting at 0. ,

..i- ,,,Now draw a number line parallel to and below the first orTe.Label the top the "number of spaces jumped." label theybottom line "number of huck's jumps" and number itaccordingly. I , . -

9 10>

-jumpedof spaces

0 I 2 3 4 5 6 ; /<

0 I 3 4

Ask the following questions;

5

Number ofdChuck's jumps

AI

HOW MANY SPACES'WILL CHUCK MOVE:IN 3 JUMPS?(Locate 3 on the "Chutk's jumps ".-line, and.look at thenumber directly above it on the "number of spaces" line.Thig'illustrates .3 x. 2 = .; however, at this time,do not writethis multiplica ibn sentonce.)

A . ,

WorksheetUnit 17 Name

Diaw bluo jumps to show 3 + 3 + 3

Draw red ;jumps to show 3 x 3 r- 9.

.,

t >2 3 4- S 6 7 8 9

:Draw blur: jumbs - 4 6.

Draw red Jumps to show 2 ic 4 i3.

Ent-- '1 t ti..)e I .:t 3 4 7 3

NOTE: Worksheet 8 showsan Australian.tree frog.

.

Worksheet 9Unit 17 Name

- !----;--"'I<I I I.I I 1'1 I i 1 I I I I I 1 >

0 -I 2 '3 .4 5 6 7 .8 9 10 11 12 13 14 15 16

Write an addition equation about the steps on thisnumber line.

. .

Can you write a multiplication equation about thesesteps?

-. 7vb .,

...

<11111111,iiiiiii_+_>0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 IS 16

A bug jumped up and came down just where he started.He jumped O unit spaces:Write the multiplication equation telling about this. -

----s

..

HINT: Think afxi.ut these qupstions.How many jumps did the bug make?How long was each jump?

After the children have completed the problems on Worksheet 8,ask them-what they have discovered about the red and blue.:markings. (They, are, of course, the same.)

i

14

F

Lesson 3: SHOWING MULTIPLICATION ON PARALLEL NUMBER DINES .

Here a parallel number line method of showing,multiplicationis introduced. We will use the jumps of the Kangaroo familyto develop two parallel number lines of different scales. Thesepairs will be called a 2-sbale or a 3-scale or a 4-Scale, etc:,It is not intended that the students learn the Meaning of theword "parallel" . It is important they learn, to read the productsfrom the number lines,

'Similar parallel scaled, number lines are developed in Lesson 4"ofNUpit 18.

MATERIALS

Worksheets 10, 11 and 12

PROCEDURE

.. .. .rDraw a number line On the chalkboard. Have one of your

students show Chuck Kangaroo's jumpS on the number linestarting, at 0. .-.\ -'

,..4..1- A,,t ,

Now draw a number line parallel to and below the first one.Label the top line "number of space's jumped." 'Label thetbottom line "number of Chuck's jumps" and number itaccordingly. ' . . '-

c^-" ThJ

C

< 4

04

I 2) .

3 4 5

I

:

6

'7

$

9

0 '.2 3 4

4

Ask the following questions:

1 > Number of spaces .10 -jumped

> Number of N.

Chuck's jumps

HOW MANY SPACES'WILL CHUCK MOVEJN 3 JUMPS?(Locate 3 on the "Chutk's jumps ". -line; and look at thenumber directly above it on the "number of spaces" line. (6)ThWillustrates .3 x, 2 = e, however, ,at this time,do not write .this multiplica ion sentoncg.)

IS

b.

A

v.

16'

,HOW MANY SPACES WILL CHUCK MOVE IN 2 JUMPS?'' (4)

Erase the arrows showing the, jumps on the top lines:

HOW MANY SPACES WILL CHUCK MOVE:

IN I JUMP ? (2)

IN 0 JUMPS? (0)

IN 4 JUMPS? (8)

IN 5 JUMPS? .('10)

Repeat the above questions, until you are certain the childrenare finding the answers from the pair of parallel lines, Callthis Pair of fines a "2-scale.",

e - .

You may suggest that one way to think of this pair of scalednumber lines is that the upper line can be "stretched" untileach part is 2 times, gs long, to`getthelower fine.

Repeat this procedure on your chalkboard by using FatherKangaroo (4-scale), Mother,Kangaroo,;(3L.scale), and KatieKangaroo (I-scale)...

After the alvve has been,completed, the thildren should do .

Worksheets 10, I I and !2. Youmay need to help the chil-dren get started, bxit entourage them totwork as independentlyai possible. The starred problem (*)'is for enrichment.

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18

Lesson 4 :' SMALL MULTIPLICTON CHAitS,:,

The tdrni "product" is introduced in this lesson. Practice is,given in recoriliqg products on charts.

MATE RIALS

,demo,nStration number line or.a number line drawn on thechalkboard

number line (0to 25) for each chile

--, Worksheets 1'3 and 14r ..PROCEDURE ..

Ask a child to,show,2 steps of ,3,unit spacei'each on thechalkboard number line.: Ha've another child write the multi-plication sentence for on the board,. (2 x 3 '6) .

Erase the marl? showing the moves onthe number line.

procedure lth other numilerso obtain the followingta

x3 = 6,

I = 4

4,x = 12 .

2 X 1 =

Ingroduce the word "product" to the class by using in yourdiscussion. When one number; is multiplied by iinother the re-sult is the prdduct.

Explain that the products can be r-scordediin a shorter way on. a multiplication chart. Drdw and explain the following chart:

2

4

6 2

12 4

$

s

g .

t.

Use sentences such as , "The product of 4 and I Is foundwhere the .4.-row intersects the I-column." Trace the rowand columnwitt your finger:: to 'show the tocation of theproduct. .

CAN YOU,FIND A CHART LIKE THIS'ONE ON WORKSHEET 13?(Yes-, in the upper left corner.)

Have the children fill in the charts on Worksheet 13. Theyshou,d use their number lines qnd work as 'independently aspossible. After they have completed WorkSheee.13, have themdo lArksheet 14.

.ot

Worksheet 13Unit 17 Naine

6

4

-.0' .3.7

2

:4

3 I .. . 1:;%

; o 0 10. /I I I I 1 I

7,

6/

0 2 3 4' 6.

A

Worksheet 14.Unit 17 Name-

Write a mJitiplication sentences for each problem.Find the product on a chart on Worksheet 13.

Tom has a train that is 2 times as loitg as Jcihn's train.John's train "is 3 feet icing. . . .How long. is Tom's train? 1

vLR...,.

5D RIM PILM 1451 t.1 lir

Bill has twice as Many balls as Tim.Tim has 3 balls.How many balls does Bill have?

x 3 *-:\ 6

Ann is 4 times as'old as Sue.Sue is 2 years old.How old is Ann?

a 1Sally walks 3 times, as far to school as Ruth does.Ruth walks '3 blocksHow far does- Sally walk?

o

C

20

Less-on 5: MAKING A MULTIPLICATION CHART USING THE NUMBER LINE

Inthisaslesson products from 0 x 0- to 6 x 6' are found 'byrepeated addition on the number line and are recorded on achart.'

MATERIALS -

- chalkboard number line - - either a.demonstratiormiumberline or one drawn on the chalkboard, numbered td 36

- Worksheets, 15 and 16

- number line to at least 25 for each child

PROCEDURE.

I. Ask a child to show 2 steps of 3 unit spaces each on the chalk-board number line. Ask someone to write both .a multiplicationand an addition sentence on the chalkboard describing the steps -on the number line. (3 + 3 = 6 and 2 x 3 = 6).

.7

Ask another child to show 3 steps,of 3 unit spaces each.Have thege equatibns written directly-below the first equa-tions.

c5

3 + 3 + 3 = 9 and ,,3x 3=

THIS TIME WE WILL TAKE JUST ONE STEP OF 3 UNITSPACES. CAN SOMEONE ,SHOW THIS ON THE NUMBERLINE AND WRITE THE EQUATIONS ABOVE THE EQUATIONSALREADY WRITTEN? (3 = 3 and I x3 = 3)

< NOW CAN SOMEONE GUESS HOW TO WRITE A MULTI-PLICATION EQUATION'WHEN NO 3-STEPS ARE TAKEN?(0 x 3-:= 0)

.WRITE THE EQUATION ABOVE THE MULTIPLICATIONEQUATION FOR I x 3.

2U

n.

, .

The chalkboard chart should now look like this:,

3

3

3 :I- 3+ 3 + 3

===

3

6

9

0 x 3 =I x 3 =2 x 3 =3-x 3 =

3

6

9

This chart could have been constructed by starting atx 3 = 0, but it.will_be less confusing for the children

to start with the more obvious 2 x 3 = 6. Continue thechart to 6 x 3 = 18. It will look like this:

0 x.3 =

:

0

3 = 3 I x 3 = 3

3 + 3 = 6 2 x 3 = 63 + 3 + 3 c= 9 3 x 3 = 9

3 + 3 + 3 + 3 = 12 4 x 3 = 123+ 3+ 3+ 3+3 = 15 5 x 3 = 15

3 +.3-1-. 3+ 3+ 3 + 3 = 18 6cx 3 =. 18

Have a child read the column of products from the chalk--lboard chart. Elicit the response that when the child readsthese-products in order he is counting by threes. It isuseful to mark the, products on-the number line. Bring outthe fact that each product can be-obtainpd13ty adding 3 tothe preceding product.

Point out to your class that the products Van be recordedshorter way on a multiplication chart such as the one on Work-sheet 15. Help the children complete the colump corresponding- .t9 multiplication of the numbers 0 ,through 6 times 3,,

tt2. In a manner similar to that above havethe following chart made

on the chalkboard:

0 x 6 06' 6 I 5( 6 = 66 + 6 = 12 ,.2x6 = 12

6 + 6 '18 3 x 6 = 18'6 21.- 6 + 6 +- 6

0=

= 24 4x6 = 246 + 6 + 6 + 6 + 6 = 30 5x 6 = 30

6 + 6 + 6 + 6 + 6 + 6 = 36 6 x 6, = 36

29 21

.)

The children should follow the board work on their own num-ber lines. until 'the produbts become greater than the numberssb.own onithe number lines. The products should be recordedon Woiksheet 15.

3. The children should now complete Worksheet 15. They mayuse the completed WorkSheet 13 ox their, individual numberlines.

4. lien the children that because it .is 'easier to find numberswhen they are.in.counting order, multiplication charts areusually made like the one on Worksheet 16. ,Have the chil-dren cut out the' columns from the chart on Worksheet 15 andpaste them over the corresponding columir on Worksheet 16.(If you wish, you may have the rows simply reA.copied inordei.) Now, both rows and columns pi .the multiplicationchart should be in order of increasing ihtegers.

DO not require the students to memorize the multiplication0 facts. Let them refer to this chart whenever it is necessary,to find a product. The patterns in the chart will bp discussedlater. How.ever,"a child who notices some of them now shouldbe encouraged in his observations.

NOTE: Have the children save Worksheet 1-6 for use in future lessons.

Work6heet 15Unit 17 Name

X 3 2 1 S 4

0 0 b 0 0 02 / s y 0 6

G -i-/ to 8 2 42,

9 6 3 18 .4 0 18

12 f zo 16 0 24

/.5: /40 ,c- C 20 O 3o

18 a 6 30 24 0 36

22

WorksheetUnit 17

X

16

0

Name

I

,

2 3 4 5 6

° ° ° 0 0 0 0S.-

061 o/

2 0 2 4 6 g` 7,5 hZ,

3 0 3 6 g 42, /5 18

4 0 y. i 12 /4 °to c,11

5 so s Jo /5.,20 .2. 5 ge)

6 a 6 /.7 a .011' go ,6.

...,

SECTION 2 ARRAYS-AND MULTIPLICATION

In this section the children find products by counting objectsin arrays.' It is shown that-this method is very closely re .

lated to using repeated jumps on a number line, and of coursethat'it gives the same results..

1.2

3 1

fc

23

P.24

Lesson 6: MULTIPLICATION-USING ARRAYS

In this lesson products are found by counting the objects inrectangular arrays.- This is shown to be the same as joiningequivSlent.sets otunit -spaces on,a number,line.

MATERIALS

. demonstration number line20 counters to place on the number line (put sticky puttyor pieces of masking tape on paper counters) ,Worksheets 17 -and 18

PROCEDURE

Put 4 counters on the olesk 6f each of 5 children.

HOW MANY CHILDREN HAVE COUNTERS? (5)

HOW MANY COUNTERS DOES EACH ONE HAVE? (4)

HOW CAN WE FIND OUT HOW MANY COUNTERS THERE AREALTOGETHER? (Count or add. If Ahild says multiply-5' x. 4, change the question to ''Can yOu.show this on thenumber. line ?")

Have each of th'e 5- children in turn' place his counters on thenumber line. In discussion bring ou't the following facts: Wejoined 5 sets of 4 counters each and found we had 20 counters.The addition equation for this is 4 + 4 + 4 + 4 It- 4 = 20. Themultiplication equation is 5 x 4 = 20. (You can think of this...equation as saying that 5 times We had 4 objectS.) Each countercorresponded to a unit space on the number line, so we alsojoined 5 sets'of 4 spaces each.

Have each of the 5 childien pick up his 4 counters. This timehave each put'down his counters as one row of a 5 x.4 array:

O 0 0' 0O 0 0 0O 0 o oO o O oO o o o

32

r's

EXplainAo the class that this is a neat, easy way tcpshow5 sets of 4 counters each. The, arrangement has 5 horizontalrows and 4 vertical columns. ,ft is called a Tectangular array./'The total number of countersin the array is 5 x 4 = 20.

Repeat the above procedure `With other small integers, if-youwish, and then proceed with Worksheets 17 and 18.

Worksheet 17Unit 17 Name

the boxes.

000O 00O 00

Number of rows

Members in each row

Members in the array

Multiplicationequation

1375

3 x3:: 9

Number of rows000-,0 a .0 0 Members in each row

O 0 0 0 Members in the.arrayO 0 0 0 0Multiplicationequation yx5:20

ANumber of rows

Members in each row

Members in the array

Multiplication'equation

Worksheet ISUnit 17 Name c.

. .Fill 'in the boxes.

000000000000000

0 0 00 00 .000 i000 ) /'..000

Number of lows I51Members in each rowifj

Number of rows

Members in each row[..

Kiultiplication equation Multiplication equation

13x 5 : /5 S x ? : /5

Finish numbering the arrays and fintsh the equations.

00100010100

.

.,.x%..

-i--

0 0 C1@POO'0®C)Ci®Ci

x 3 '-= a)

3325

e

Lesson 7: MAKING ARRAYS

More practice is given here in making arrays to illustrateproducts.

. MATERIALS

individ ual pegboards with pegs , or 25 counters per child

flannel board with sets of objects (optional)

Worksheets 19 and 20

PROCEDURE

The problems on Worksheet 19 can be solved by making ap-propriate arrays and counting the-menibers . Suggest that theclass make the arrays on a `pegboard or -with counters. Youmay wish to have volunteers show their arrays on a flannelboard. perhaps some children wopld like to make up more,entries for the parade.

The terms "row" and"6olumn" In ay need tobe reviewed. Sometimes we use the phrase"number in each row"in Rlace of the equiva-lent "number pf coluMns.

The game on the nextpage may be played afterWorksheet 19 has beencompleted.

-26

It

3

Worksheet 19Unit 1? Name

%,4

Write a multiplication equation tor 'each problem.,

A band leads a narade.There are 5. rows of players in the band.There are 5 columns of playets in the band.How many players are in the band?

Some scouts carry flags in the parade.There are z rows of scouts.There are 3 columns of scouts.How many scouts are there9

Next are some men on horses.There are rows of horses.There are 3 columns of horses.How many horses are there',

I) Array Game

This game is described for an individual player. However,. itmay be played by partners or teams. The game ma also bevaried by using pegs and pegboards in place-of tiles.\\

A player takes & number of square tiles the number is deter-mined by choice, by drawing a slip of numbered paper, or bytaking a handful of tiles,. He makes as many different rectangu-lar arrays as.he can from the tiles, but-. he must use every tileeach time, The results are recorded on Worksheet- 20, Or on asimilar record sheet.- For example, from 15 tiles, a playercould make arrays of: I x 15; 3 x-5., 5 x 3, and 15 x I,.

The players should soon seesthat the arrays occur in pairssuch as 2 x 3 and 3 x 2, They should also-discover thatmany more rectangular arrays can be made from certain 'num-bers of tiles than from others. For example, eight, differentarrays ,can be made from 24 tiles an only two. arrays canbe made from 23 tOs.

3,D

The players may deyise theirown.scoring rules. Ohe po,s,sible way to score is to giv.eone point for each array and tototal the points for five trials.,The winner' is the one with Elie'greatest score.

27

\-\

se

28

..6

Lesson 8: COMMUTATIVITY OF MULTIPLICATION

In this lesson the commutative (or order) principle of multipli-cation .is presented. This principle states that the factorsmay be taken in either order; for'example, 3 x,4 = 4 x 3.The children should understaod and use the commutative prin-

.,ciple, but they neednot use the term.

MATERIALS.

2 flannel boards or 2 bdards, pieces orcardboard,or magazines ...

- 24 shapes for the boards -- preferably fish (there isa pattern at the bad: of this manual).

Worksheet 21

PROCEDURE

.Place two boards before the class. (Or place twoboards, pieces of cardboard or magazines on a table.) Haveavailable 24 'simpl fish shapes. Read or tell the followingto the children:

Seven children, on a Qicnic with their parents, de-cided to go fishing. Th- 3 boys went in one boat.

3v

1

and the 4 girls in another boat. When they came in .

for lunch, the boys Sho1.tted tthe girls,' "We have'more'fish than you do. Each of ifs caught 4 fish."The girls called; "You do not; Each of us caught3 fish, and there are more Of us." Who was right?

Place the 4 fish .for one boy in -a row on one flannel board(Or on one of the boards or magazines) saying that you aregoing to make a picture of the ritirnbet of fish _caught by thebbys: Place under this row the rows of fish f6r the othertwo boys, producing ,a rectangular array.

. .

Ask a child to count the total number of fish and write thecorresponding multiplication equation on the blackboard.(3x4 = 12, not 4 x 3 = 12)

Similarly on the other flannel board (or board or magazine)Make a rectangular array showing the girls' fish.

.

Again have a child count the total number of fish and write the.corresponding multiplication equation. (This time, 4 x 3 = 1'2)

Someone should by, now have volunteered the inforination thatthe boys and girls caught the same number of hill. If not,bring this out, and then ask ifithe arrays for the boys andgirls look alike. A child will probably suggest rotating onearray to produce the other array. Actually rotate one of the

J

O

'29

flannel boards to show this. It does not matter whether wehave 3`,Yows of 4 fish or 4 rows of .3 fish; the total is the same :3 x 4 = 4'x3.

The children Should now be ready for Worksheet 21 where more.examples of the commutativity of multiplication are giyen.

I,

Worksheet 21Unit 17

Finish numberingFinishthe equations.

_

Name

the arrays.

O.

2 4( 3

G ® p

:, 6 .

"o

0 @3 x ,7i=

, .E/ 3 # 5a7 9 /0 E 2-

Ea a /3 7# /0 El a

4_ x 5 = /5/1 1: 1 4E

5 x j : /5-

30

38o

.

Lesson 9: MAKING A MULTIPLICATION CHART USING ARRAYS'

In this lesson; as in Lessbn 547 a.- multiplication chart ismade-up. to 6 x 6. MIS time thelproducts are found by usinearrayS.This method differSfrom,. but is 'closely related to, the methodusing repeated addition on the number line. The charts will. beSeen to be .identical.

MATERIALS .

Worksheets 22, 23-, 24 and 25

PROCEDURE 1

oe

Tell the class that they will make another multiplication chartby counting the number of members in arrays.s,an example,ask a child to and count a 3 x 5 array.

0 0 0 0 0o o o'o o00000

All the children should enter the product 15 in,the box Wherethe 3,-row and the 5-column intersect on Worksheet 25.

Ask if they should:draw a 5 x 3 'array to find the product5 x 3. If the answer is no, agree and point out that,only half

thb products in the chart need to be found -because the otherhalf ju. st.have the order of the. factors reversed. (Note thatthis is the first use of the term "factor." Introduce it casuallyby repeated exaMpre.). If the answer above is agree that.this would give the product 5 x 3, but ask if there is an' easierway. This should bring^a suggestion to chahge the order of the'factors. Someone may also' correctly suggest viewing the 3 x 5'array from the side. Have the children enter the product 15 inthe bOx where l -row and 3-column intersect.'.

Now tell the class that you know another-short cut for findingthe products. (This "short cut" is primarily introduced to givepractice in recognizing relations among numbers.) Instead of

0

31

9

.1

O

32

drawing ail then counting arrays', wecan dumberthe members as we go along.

Have each child complete numbering'thP'top row of the first 'array on Work-sheet 22. This numbering slows' that.I x 4 T- 4.1 Have him 'Write this equa-tion.beside the array.. \

Continue by having each child` numberthe next .roW'of four-boxes and writethe appropriate equation. Continuebuilding up the array until it has sixrows and 24 oxes.

dr!

L

I

I 2 3 4

5 6 .7 '8

'9. 10 11 12

13, 14 15 16

17 18' 19 20

'121 22 23 24

4

2 x 4 '8'3 x 4 = 12

4*-='16

5 x 4,= 20-

.6,x 4 = 24

Discuss this array. in ciaAss. One fact that should be brought:,out-is that each pIoduct can be obtained by adding 4 to the

rfi

Worksheet 22 nUnit t7 Name

156

9401/ 42

Ix2=x.?

x 2 =

.4 I x; 7 _2_

x 2 = / 06 x 2 =

Worksheet 23Una 17 P Name

I 2 '3 4g9 p

a /2. /3 /ei'16 17 a'

z,.1 0 ?? ai 075

.ge, .?7 28 29 3o

1 x =

otx 5 =La_IX 5 =

5 = ptio

.6--x 5 = otC.Ax 5 =

3

57

7 9

/41

///

14

16 ,17

.1 x 3 = 3.2 ).x 3 = 6.

3x31 .35x3 = tc'

,.:Lx 3 =

. ,

preceding one, hence reading the products in order is countingby fours.

. .

Enter the products in the chart on Worksheet.25 in the 4-column.Also complete the 4-rOw by using The commutative principle.(The children will notute the word, just the idea.).

'Have the children complete Worksheets 22, 23, 24 and 25. Forproducts where one factor is zero, explain that, for example, .

0 x 4 can be thought of as the number of members in in arrayof zero rows.of 4 each, whicll contains no members.. Similarly,one carnhink of 5 x 0 as the number of members in an array of4

5 rows each containing zero members.

/Whe.n the chart on'Worksheet 25 is completed, have the.chijdrencomPare it with Workshee;16. If no errors have been made the

. charts will be. identical. Although the mechanics of obtainingthe charts are quite different, the underlying concept in both

Worksheet 24unit 17 Name

6- 67 e ?Bill!0 Az 1 s 419 g 0 i:Z/ .

Ec?1

25 SI 27 28, 9 w3/ 3:? 33 ga 35 Eil

4

33

4 "\34

O

methods is the use of repeated addition to obtain produgts ofcertain positive integers. This fact should be discussed with.your class to the extent you judge worthwhile.

Have each child save his copy of Worksheet 16 or Worksheet25 for future use.

Clap GameThis game may be. played during or after Lesson 9. Ask the chil-dren to count aloud and clap on the multiplei of three (for exam-ple). TheYwould say, "One, two, three (clap), four, five six(clap)," etc. **

Buz Game,This is a more difficult version of the above. Ask the childrento count al'oud, saying "Buzz" ,instead of the multiples of three(for ex.ample) They would say, "Ole, .two, buzz, four, five,buzz," etc.

4 2

Lesson 10:- AREAS OF RECTANGLES WITH INTEGRAL LENGTHS

The concept of area has been developed in kindel-garten andfirst grade units. Here we use the number of square unitsincluded in a rectangle with sides,of integral length as a mean-sure of the area included in the rectangle. We do not developthe rule, area equals length times width.

This leSson, uses the idea of a floor plan. The childred2willbe familiar with this idea from their lessons in Scaling andRepresentation. '

MATERIALS

-for each child --

about 20 squares (e.g. , unit Minnebars,square counters , tiles )

- Minnebars- Worksheets 26, 27, 28'and

PROCEDURE

Read or tell the .following-to the children:

square blocks,

The men in the Australian machine testing,camp Wereconstructing a temporary building: They planned tomake the floor of square pieces of plywood and theywanted to know how many plywood squareg they wouldneed. Bill measured the floor. He found that exactly3 squares fit across the end of the floor and exactly 4squares- fit along the side of the floor. A small-scalefloor plan showing this would look like this:

4 3

(Sketch on board.)

O

35

.

. o

)

, a

36

it

i

I

0

Ask-each child to use his squares to-make a plan of thefloor showing how the plywood squares would cover it. , I/

HOW MANY ROWS ARE THERE? (3)

HOW MANY SQUARES ARE IN EACH ROW? (4)

HOW MANY SQUARES ARE THERE ALTOGETHER? ( I )'i

WRITE A MULTIPLICA'HON EQUATION THAT TELLSTHE NUMBER OF SQUARES . (3 x 4 = 12)

Tell the class that the number f squares is a measure of thearea of the floor:. .Repeat the above activity with other counting:numbers , if youwish, before going on to the worksheets.- SuggQst that the-children check their answers on Worksheet 26 by fitting,Minne-bars onto the diagrams and counting the unit squares.

. ,.

4.

44

9 '

4

Worksheet 26Wilt 17 Name.e

Number of rows

Number of _columns

This Is a x _6' rectangle.The area is /..5- squares.

Number of rows 6

Number of columns 4.4

This is a 6 x rectangle.The area is squares.

Number of rows .6

Number of columns

This is a k rectangle.The area is squares.

Workshe'etUnit 17 Name

San.ly wants to the the floor of her playhouse.She placed tiles aloTig 2 sides of the floor.'You shOw the tiles by plqcing Minnebars on thepicture of the floor.

:low many files will she-need for the whole floor'Wrde a tr4Itiplication sentence telling the numberof tiles.

/

.. A border of a flower garden is made of square 'blocks.Here is a scaled-down picture.

The area of the border is square blocks.

The area of the garden is 0 square blocks.

Worksheet 27Unit 17

.

Place MinnebarsWrite a multiplication

Narrie .

t,Inside the rectangle to find-the area..

sentence about the area.' ..-

2,( 6 :.- aii

-1" 4 4- -1- 2 -i. I. , 1 1

-I- --1

- -- 4 --1--1

I I - I 1 I.

g 3 . 6 '1

!

I .i 1

I 5,r 5 : .2,5

1 1 1

1 1

--+Li--t*-4--I. . i 1 1

--+1 1 "-r T' --t --1.7. 4 --I

I

4

Worksheet 29Unit 17 Name

Billy had a 4 x 2 rectangular array of blocks. Fredhad onehat was 4 x 3.,*George had one that was4 x 5.

11111 111111 MINN11111 MUMIN MIMEEE SUM

Fred's George's

How many blocks did Billy have?

How many blocks did Fred have?

k. .

How many blocks did George have?

If Feed and Billy put their blocks together, would theyhaVe more blocks than George?,

37

SECTION 3 COMPARING ADDITION AND MULTIPLICATION

Addition and multiplication are twd different mathematicaloperations. This may be difficult for the children to seeuntil they are able to find products by an algorithm that doesnot directly involve repeated addition.. However, this 'sectiongives the children a start toward understanding the conceptby comparing the results of adding and multiplying the wholenumbers from 0 to 6,

r

'vs

38

I

Ldsson I I: MAKING..AN ADDITION CHART

In this lesson the children are asked to complete a 6 x 6addition chart. This chart will be used in the next lesson.Making the chart will also provide a review of ways of find-ing sums.

MATERIALS

- Worksheet 3,0

- addition 'aids' such as number lines, counters , and additionSlide rules

PROCtDURE

Have each.child Complete Worksheet 30, which is an additionchart with sums from 0 + 0 to 6 + 6. There will probably bechildren who fill. in the chart by memory or by a simple count-ing procedure. However, .it is expected that there will beother children who will need to use aids such-as number lines,counters, and addition slide rules. Discuss with them theidea that only half of the sums need to be determined and theother half can be recorded by using the commutative principle.(However, don't use the word with the 'c)-iildren). For example,after determining that 5 + 6 = I I , one immediately knowsthat 6 + 5 = II.

Do not require theclass to memorizethis chart. 'The ob-jective here is tohave the chart cor-rectly completed sothat it can be us.edin the next lesson.

WorksheetUnit 17

+ 1--iL.__

° :IH,

t

i--

2

30

0

Name

1 -4;" 2 3 41

5 1 6'---

7 oZ s_ 4 ..5- 6

/;2u

,7_,

5

5 1(.5-

5

6

6 74 7 5

3 11 ,..5- 6 7 g ?g .5- 6 7 8 g /0

5 .5 6 7? 9 /0 /7, 6 7 8.9 70 / 7 12

.

39

\ a.

40

A

Lesson 12: COMPARING ADDITION AND MULTIF:LICATION.

A comparison of a Multiplication chart and an addition chartis made in this lesson.

MATERIALS

completed copies of, Worksheet 30 and Worksheet 16 or 25

PROCEDURE'

I. Each child should have on his desk an addition and a multipli-cation chart. Ask questions such as these:

DOES 'IT MATTER WHETHER YOU USE WQRKSItIEET.'16 OR 25FOR YOUR, MULTIPLICATION CHART? (No, we saw they arethe same.7.

ARE THE NUMBERS ON THE TWO CHARTS ON YOUR DESK THESAME? , (No.)

'nf WHAT IS THE NUMBER IN THE.2-ROW AND 4-COLUMN INYOUR ADDITION CHART? (6) ",WHAT IS ANOTHER NAME FOR IT? _i(2 + 4)

WILL SOMEONE .WRITE THE ADDITION EQUATION ON THEBOARD? (2 + 4 = 6 Dr 6. = 2 + 4)

WHAT IS THE.,NUMBEil IN THE 2.-ROW and 4-COLUMNYOUR MULTIPLICATION CHART? (8).

WHAT IS ANOTHER NAME FOR IT? .(2 x 4)

WILL SOMEONE WRITE THE MULTIPLICATION, EQUATION ONTHE BOARD? (2 x 4 = 8 or -8 = 2 x 4) /..

Point out the different symbols' used for the two operations andthe different.r6suls of the operations.' If you wish, lepeat Thequestions with other numbers.

'7

'

14

2. Have the children-look for patterns .in the charts. They should,notice' some of the ,following patterns,, although they may notdescribe them in these words.- They may notice other 'patternsalso.

,Each, chart is symmetric about the diigonal that goes fromtheupper left to the lower right because both operations are comrmutative. In the multiplication- chart the rows and Columnsinvolve 'skip counting"; for example, the 3-column reads 0,3, 6, 9, 12, 15,, 18 where each term is foUnd by adding. 3to the preceding ierm. The "terms in a row or column in theaddition chart .dre.in counting order; for example, the '3- columnreads 3, 4, ^5; 8, 7,A, 9.- The ascending diagonals on the.addition chart have identical Wins., the descending diagonals' areeither odsd or even numbers in order.

41

ro

ti

5

JIMMYSFAVUIIITt COOKIES

jimmy came runniKX6fhe from school, and rushed into the. , , ...---

house. "Mother, Other!" 'he.called. "Mothei,.w.here :are you?",..

.,

"Here I am; \ ,ijimmy, i his mother called from the kitchen;"What do you want to tel.-n-4e"

Jimmy ran into the kitchen, ";Mother, some of the boys in my

5

O.;

1

I

(c) Lite a 5-scale to show .4 x 5 = 20.

(d) Add the number of books directly as 5 + 5 + 5 = 20or 4 x 5 = 20. A

Some children, immediately on reading the problem, say or--write 4 x 5 = ,20. Naturally 'this is very good, but,-aik themto illustrate the solution with a diagramthirtime and a fewother times.

Neither the numbeffines.nor the arrays need to be gre-ciselydrawn with rulers.

Worksheet 31Unit 17 Name

. A bookshelf has 4 shelves.

There are 5 books on each shelf.How many books are there?

9x5.20Mary hasa garden with one row of beand,

erow of peas, and one rotv of corn.Th e are '6 plants in,each row. ,

How any plants does she have?

.. John's room h acwindow with small panes. .

<1.

There are 4 ro of 3 panes each..How many`panes a e there?

. ,

51

worksheet 32Unit 17 Name

I. Bill measured the length of the table with a6 inch ruler.

He found the table was as long as 4 ruler lengths.How many inches long vk.s. the table?

5Ix =4.2.54

2. -lane poured 3 pans ofmilk into a large bowl.

Each pan contained 2 cups of milk.

How many cups of.milk.were in the bowl?

3. Three cats each had 4 kittens.How many kittens were there altogether?

?S

Worksheets 33 and 34

Soine of the' problemson these worksheets are answered .byan addition equation and .some by a multiplication or repeatedaddition equation. If a .child correctly uses a repeated ad-dition equation, approve it, but ask if he'can also write the* .

corresponding 'multiplication equation. Deciding which opera-tion to use may be difficult for some children, but it is basicto the understanding of the operations,

WOrksheet 33Unit 17 %am°

074.A.I`store tias 4 red incycies aria 3 blue bicycles.

Flow rany red and blue ID cycIe does It

\ 2. The stoiehas 3 bicycle Naas'.Cacli'can hic.A bicychys.

How many bicycles can the racks hold'

\ 3 /-3. A truck brings nownew recrbicy es and 3 new

green bicycles to the store.,.

flowmany new bicyclest,cfoes it-bring?

44 a

5

a

3;:rlt 17 t's

I . 9 toys ea1.h. n

How riany'r;ttbirs ,10 they have alto ether'

4:2' 0

2. Tor, hls cars.Salt hak toy

flow mhy da Tort, and Bill have'to;;othee'

6 it at

3. Puzzle Pr- oblem:' If it takes two minutes to make

each Cat, licrut long will it take to -cut ti-lootr pole into` a1 equal pieces'

, S.

1.

?"02,2z-es

"

J

SOME FRACTIONS. ...... .. . .. -

This section presents some simple. fractions. 'Children have.practical use' for fractions, at this age', In additiOn, fractions

"'provide a bac4grourid.fOr the development-Of,,divissiori.in laterunits.

3

The' children establish familiaiity with fractions both by divid-ing a set into equivalenissubSets:arid.by finding fraotiong partS. .. ,. is

*f geometric _figures . For example, they will find of 12 by i

. .. ,dividing a set of 12 members into three subsets of 4 menibers.-.. .

-each. In another example they will find'8 of the area of a rec-

tangulartangular shape by folding the shape. into three equal parts.

I

ta

a

45

1*

0.

Au,

Lesson I4: SO E SIMPLE FRACTIONS

(

*4` V

This4esson Introduces simple fractions,' such as one-half,'one - third, a d one-fourth, by considering the separation ofa set into a Itl* opubsets containing the same numberof Members, t at is, into equivalent subsets. Far example,

there are thr e equivalent subsets, each is one-third ofthe original set and we say that the number of elements ineach subset is d-third';of the number in the original..

The 'children shou d be encouraged to develdp their ownfnethods for breaki g a set into equivalent subsets.,

MATERIALS'- .

. a flannel-board with 24 objects 4epresenting cookies

counters optional)Worksheets 35, 36 v '38 and 39 .

PROCEDURE

Begin by reading the star/ "JimniY-'s Favorite .Cookies, 'Lpausing for Class participation as indicated, The variou sdivisions of 12' may be indicated with flannel' board items ,or with-counters, or perhaps with drawings on the -chalk:,board. It is ;recommended that you always write a,fraCtionwith 'a horizontal line rather than"ivithoa slantect*line, e:g.1

at

46

t

4

raper than -1/2,' because the form mithtlie horizontal

13:n9/is easier far children to read and Write.1

5 4

I

fr

a

i -JIMMYSFAVOITt CO,OZIES

.

Jimmy came runnilugth:othe from school, and rushed into thehouse. "Mother, Mi6therili "Mothei,.w.here are you'?"

" liere I aril; 'Jimmy,') his mother called from the kitchen;"Wilk do yOu want. to tel 4110?"

Jimmy ran into th itChen. 'Mother, some of the boys in thy

5o

.

room are going on a hike this afternoon, right away._ was wdneler-

along something to eat. We're supposed to meet

hous.e, at:3 o'clock."ing if I could to

in front q

ji my's mother thought for a minute. "I don't know what I

could give you. I haven't been to a store for almost a week."

bat about apples?" suggested Jimmy.

'N.6, you ate our last apple yesterday." Her face brightened.

"Oh., es, I know just the thing." She reached up to a high shelf,'and rought out a white paper bag., "Mrs. Brown stopped in this

mo ing, and left us a dozen peanut butter cookies. Now,- I don't

wa t you tosbe eatii.g all of these before supper, but you can taKe

. th m on one condition."

"What is that condition ?" asked Jimmy, a little uneasily.

"You must divide these twelve cookies evenly among the boys

Who go on the hike ,*".. said Jimmy's mother./

"You had me worried," Jimmy replied. "That isn't such a

bad condition. I promise that I will divide the. cookies even,

among all the boys who come on the hike. I was expecting soigne-\

thing much worse, like having to weed the garden, or scrub the\I

.

porch." , \

.

,

\t

i Jimmy's mother laughed. "'No, 'I won't ke advantage of \'

yo r wild craving for peanut butter cookies. Now it's just aboLt.

1

3 o'clock. ,YOu'd better get out in front, and meet the they boys., f ' . ,

ere \are the cookies. Be sure to be home oefore,supperti e and,,,

have a good hi e.

"'Goodbye, Mother, "prid thank you," jimMy called, as he w

out the door.5 0

\

,4

;,

Jimmy waited in the front yard.-t,He was looking forward to,the hike, buiright now he was thin,king about the 'peanut butter'cookies

.in the bag he was holding. He-was getting hungrier and 'hungrier justthinking about them., QcCassionally" a delicious smell of peanut buttercookies would find its way out of Vie, bag and float uPward, to his \,;.

nose. Most mouth-watering of 'all smells!

At five minutes after three,' none of the ,boys in his- schoolroom ,

arrived.' jimmy was somewhat disappointed, and even impatient:But he. had another thought. "If 'no one shows up for this will.hcir( all twoiv.0 of the peanut butter cookies for myself."

Bert at six minutes'after three, before Jimmy had had too

think about eating all twelve of the cookies, his friendI' anal arr,.ed, all ready fir a, good hike. jimmy greeted Pala, bi..,,,f;e

WI ,dv) t!unking, ".We'll have to divide,the twelve cookies evenly.

,,lares. Paul and I W,111 each got one -half of the _cookies,w will. each get --="

MA's;Y COOKIES WILL CACI' BOX. GET? If a set4 1.: thing:-,, 1:: divided evenly 'into two subsets., each sub=

ki.ontain things. We say a subset of B thingsOf a set ref 12 things, and 'we also say 6 is .

Junrly was, thinking that e could yeallY fill up pretty well withpcanut: idutIor cookieS and hea was 'glad te have Paul's comarqy

.Rut at that moment theiNfriend John came into the yard,

",-)h-,,oh," thought Jimmy, "Now we have to' divide thec(,) ,Ienly into three Shares That' means at Paul and John

I will c:,,ach (let one-third. \Ye will cachget

57

4r

50

aCHOW MANY COOKIES WILL EACH BOY GET NOW? (Your.Stress the fact that if a set is divided evenly into threesubsets, each subset represents one-third of the total,

and 4 is3

of 12.)

BefOre jimmy could become completely used to the idea of

having only four cookies , Henry came into the yard. Jimmy tried to

hide his disappointment. He tried to.act happy to see Henry arrive.

But he was thinking, "NOw, we'll have to divide the cookies evenly

into four shares. We will each get one-fourth of the cookies. That

means we'll each have ---"1( 4 of 12. is 3. Develop a discussion similar to the

firevibus. one )...

Jimmy loved peanut butter cookies, but he was also a good

sport, and he was able to enjoy the boys gathered in his yard, eventhough he would now get only three cookies. And just then the twin

brothers, Ed and Ned, came running up to,the group.' Jimmy thought\,to hithsel"Now there are six of us.. That is all who said they could

come. That means we'll each,.get one -sixth of the cookies. And one-,

sixth of.twelve is --;"

(6

of 12 is 2. Develop a discussion similar to the

previous one".,)1

By now it was quite clear to JiMmy that he.would get no more

than two of the cookies he was carrying but he made a big effort,

and smiled as he greeted Ed and Ned.

'Just as the boys were leaving'the yard to begin their hike,

Jnilmy noticed that Ed and Ned were each carrying a white bag.

"What do you have there?" he asked.

J0

o

rL

,Ed and Ned answered togetligr, "_Our mother gave-us eachsix pearaffbutter Cookies/to bring al9ng. But we have to dividethem evenly among all the boys on the hike.:

./zNOW. HOW/MANY COOKIES WILL EACH BOY GET?

I,/

b(Four. of 24 -is 4.)

Worksheet 35Unit 17

Have the children do Worksheets 35, 36, 37, .38 and 9.

Here are I0 trees.

The trees are divided into subsets.

LA6.

Here are 141 triangles.

The triangles are divided into 3 subsets.

There are

61bleach subset.

is 1

of 18.

Worksheet 36Unit 17 Name

GOO00eHete are cookies.

2 boys will share them.

1Each will get of 6.

of 6.

Jane .had tulips.

She ,planted them in 3 rows.

1Each row has3

of 12. .

1is3

of 12.

51

dVorksheet 37Unit 17

;ere Jre'' .Itten.Fr.

ity'rle the zattew tntet,.vo tubr.:et .1 the ;.-vne nu-nber.

Eych of the two

of II!) is

,trosets kittervn.

5 .

Here are tr:rtles.

Dr.fw- cur., dr.lle the turtle: .

group-;. 6: the = r^e number.

Each ot the 4 'ritoup.- ha; tattle

of t 2 I,

Worksheet 38Unit I7 Nome

, .

.

*00001.

Color of this set.3 .

I

-i of S Is _/ .

N N ,

Si A 11k

I

Color -, this ,..z. t.

1

of f.: s:

* CV * ® ($),00000.

0 o aoo.

o o 00c..

'Color 4-'-of this set.'

t of 26 is .:5 :

.

Ch

0Co or of thts set.

.:

1

f b

4

Worksheet 39Unit 17; ,r Name

.

0 0COO

51

..i.

.

L.)1 f 8 t', 0 s .

.

---.

t of 6 it;-3-

. .7

:

.

0' .0: 00 0 0..0000 .0" 0

1 of 12 is3

.

Z .LALA ,A

A A

i-'4

of -12 is

.

.

AA L'

$.' ,..'...,_.,....

.

\-

Lesson 15: FRACTIONAL PARTS OF PLANE FIGURESN

This lesson gives another illustration of the meaning of cer-tain simple fractions. A.whole object is separated into 2equivalent parts (hales), or 3 equivalent parts (thirds)., or4 equivalent parts (fourths).

MATERIALS

-7- scissors ,

Worksheets 40, 41 and 42

PROCEDURE.

HaVe the children cut out the three shapes on Worksheet 40.Have each child pick up his circular piece.

CAN YOU. FOLD YOUR CIRCULAR PIECE INTO TWO PARTSOF EQUAL AREAS?

Fold your piece into,-" *

halves along 1,vith<t1ie.children. Show' themthat one partlits ex-actly p.ver,the other. .

Dtplain' that each parthas 1 of the total

2

area. After:the children have completedthe folding, have 01-erndraw a pencil linetracing the fold of thecircle. Label each

part 1 . Then have

nem fold each of theother shapes in half,trace the folds andlabel each part.

Vor'LshL.,t lc'nit 17

L

Cut Out tho 5haPeS.

r0 the shapes into 71.3.

Trace Ult. folds With your pencil.

Wr,te 7, on each part.

6 i 5.3

a.

Do Worksheet 41 in a similar;/ anner. Some children may heInterested in using other fp51:Is nd other shapes. You Might.suggest trying t fold a ycle'i to thirds.

, A /

Have the children complet

WOtksheet 41.,Unit 17

Cut out the shapes.,

Fold the circular shape into

Write o4

n each part.

Fold the rectangular shape into 3 '8,Write

3on each part,

4

Works eet 42.

,Worksheet 4Unit 17 Name

The shapes below are divided into parts.

Color only thel shapes where the parts are labeledcorrectly.

z

r

1

54 0

Lessoh 16: HALVES ON THE NUMBER LINE

In this lesson points halfway between counting numbers arelocated and' labeled on the number line.

MATERIALS

three, strips of paper about 18 inches long

Worksheet,0

PROCEDURE

.

Draw a diagram similar to the following on the chalkboard.The distance between 0 .and I should be about 18 inches.

O

0

Ask the class to suggest ways to find one-half of the unitspace betweeh '0 and 1 . One way is the following method,but they may prefer another method. Cut a-strip of paper tofit 13.etween 0 and I oh the number line, fold the strip inhalf,.open it, and use the crease to mark the I

point on thenumber line.

Now draw this segment of the number line.

0 I 2

Have the class'find the point halfway between . I and*, 2.

Tell the children that this point ig2

unit space farther alongthe number line than I ; it\is I and

1

spaces,:froin 9., LabelIthe point I and. 2' Then show them that a shorter Inlay to

1write this is 1 2 .

6 3

55

Draw this segment of the number line.

4

I,

Have the class find the point4' \

',Agdin.l.abel the point with "4

Have the children cbmplete thTubber lines on Worksheet

halfway between 4 and 5.and I ,, and with 4 .

e labeling of the pointS on the

p1

Worksheet 3.Unit I7 Narrc-,,

rtrush naminZI the p..)ents.

0 .r;-.2.4

0

I

t,3

< I

8

., 56 . 64

..

. ,.#

MOM REVIEW

A

- 'When a child has completed the fi t four sections of this unitlie should be able to perform theta s listed below, although .

his perforMance on the fourth item m y be imperfect. Only thpintegers 0 through 6 need be used p the multiplication pro-blems.

1. Interpret multiplication as repeated a dition.2. ShoW multiplication on pargllel 'numbe lines of

different scales.3. Make rectangular arrays -and use them to interpret

mUltiplication.4. Differentiate between the operations of addition an

multiplication and usethem correctly in si piesituations.

1

1

1 1\5. Interpret the fractions 1

,4

and6---5 -

-4

dividing sets into equivalent subsets and areas intoequal parts.

The items itithe list are'reviewed and extended in Un t 20.However, during the-period between Unit 17 and Unit 20, youmay wish to reinfo1ce the Children's ,concepts, by using theReview Worksheets 44' through 51.

57

Worksheet 44Unit 17 Name

Finish each equation.Show each problem on a num.er line.

i.---f*_.. ./ -../0., 1 .

3 x 2= a..

o .

''.:.,., ...

,6 '

..

. 1

. -' -'-----e .The0' I .

3 x 3 = 9

. .

0, 1 2 3 4 S. 6 7

. , 1x 13 = 4.5

9 10 II 12 13 14

,4

Worksheet 46 **..,

nlinit 17 Name

,:f 1 , 4

Color3

of these flowers. '

SO a I I PP.o

Wi; '1.00 'Irr iPirr IP.III _III 4 tiell Tip Is Illi el

11, 'I ,...SO It 1

..

.

It

Color of these leaves. 14,.

4

.. 0 0 0 0 .

0 q 40

0 0 4,

'Color of thesellowers.3 .

V 0.

° ..

. 0 .

Wpiltsteet 45Unit 17 Name

brawa 3 x 6 rectangular array.

The array has members.

..This shows that 3 x 6 -= IF

X Xx x xx )4. X X, X Xx X X .X

Draw a 6 x 3 rectanigu ararray.:

The array hasI (' members.

This shows that 6 x 3 =-I'Al

S

Worksheet 47Unit 17 Name

Finish-these charts.Use your counters to help you.

3.

S

2

6

4.

go

Write the'equation that nswers this problem.Mark the answer on one, the charts.

'Bill has 3 'boxes of lkoks.Each box has 4 boo4 inHow many books doesile have?

3 x 1/- /2,

J

V

Worksheet 48Unit,17 Narne

I ;Write the equation that answers each qbestion.Use your numberrlines or counters to help you.

-

I "Sam,, Ted, arieSally areinaking a rock collection.Ted brought 42' rocks.

, Sam brought rocks., .

How many Kicks did they bring together?r , A

2. Sally also brought her rockS.

The'y pul all the rocks on the table.

They made S rows of rocks.

Each row had 5 rocks .

How many rocks were on the table?'

4

,Worksheet 50,Unit 17 Name

- 4 1.1 '3 t t .1 1>t) 2 3 4,, ',5 6 7. 9 19 11.12 13

Write an addition equationabout the steps on thisnumber line.

,Write a multiplication equation about the same steps.A :4/ x 3

<1 I I '1 1 1

0 I 2 3 4 5 6 7 8 9 10 fl 12 13

Write an adcHtic5d equatibn about the steps on thisnumber

Write a mult plioatiori equation about the same steps.

I

C

I

Worksheet 49ghit 17

At Name

Write In the missing numerals. ..

O O

.)

O

t)!os

a.0

0CZ.:

-a.to

oC C0IA el

t7 0

CR

1

X

Worksheet SIUnit 17 Name

Write the missing numerals..

I o2 g 5 .67

.

/0 t.2 iS /4zk /6 '17i3I. 49

44 ,7 23 ^01 4/ 25 26 27 .g g

x 7 =

-2 x 7 ='.p X 7 = alt -4 x 7 =

o.

S9

-Fat

her

Kan

garo

o4

k.

Joey

, Kan

garo

o-

ic

Kat

ie K

ang.

aroo

Tr

6Q

Kangatoo Cutouts for Lesson 1

.63

Fish pattern for Lesson 8